On the non-dominance of counterions in the 1:z planar electrical double layer of point-ions

Abstract

According to the dominance prescription of point-ions in the non-linear Poisson-Boltzmann theory, proposed by Valleau and Torrie almost 40 years ago, the microscopic and thermodynamic properties of an asymmetric binary electrolyte converge asymptotically to those of a completely symmetric electrolyte, in the limit of an infinite surface charge density of a planar electrode, if the properties of the counterions are the same in both instances. By using the Grahame equation and the non-linear Poisson-Boltzmann theory, we show here that this prescription is certainly exact for the mean electrostatic potential at the electrode's surface and for the capacitive compactness. Contrastingly, analytical and numerical solutions of the non-linear Poisson-Boltzmann equation show that, in the limit of an infinite surface charge density of the planar electrode, it is possible to observe finite differences between the local mean electrostatic potentials and electric fields associated to a 1:1 and a 1:z electrolyte at places near the electrode's surface. Thus, we prove here that even in the absence of ion correlations and ionic excluded volume effects, the counterions do not fully dominate the structural properties in the entire electrical double layer in the non-linear Poisson-Boltzmann picture, which is confirmed through comparisons with new Monte Carlo simulations.